This episode is transparently a riff on The Taming of the Shrew, with a bit of Helen of Troy thrown in.*

But as I was watching it this time, I got a very different vibe in several of the scenes. It helped that Ambassador Petri really does bear some resemblance to Claude Rains's Captain Renault. In a green-skinned alien kind of way.

* Yeah, "Elaan of Troyious"? Really? It helps even less to know that the original title submitted with the script of this episode was "Helen of Troyius", and it was changed to Elaan to try to disguise it.

Transcript

1
Captain's log, Stardate 4372.5. Escorting the Dohlman of Elas to neighbouring planet Troyius to secure peace by an arranged marriage. Ambassador Petri is to instruct her in Troyian customs.
Kirk: Energise.
2
Elaan: I am Elaan, Dohlman of Elas.
Kirk: Greetings, your-
Elaan: I did not give to permission to speak.
Caption: Her guards are glam rockers.
3
{Uhura's quarters, temporary lodging for Elaan}
Elaan: I stabbed Petri, the Troyian pig! I hate Troyians and their "manners".
4
{sick bay}
McCoy: Petri will survive. But you have a problem with this woman, Jim.
Kirk: Of all the starships, in all the Galaxy, in all of the universe, she walks into mine.
5
Chapel: What do men see in these uncouth Elasian women?
Petri: The merest touch of their tears is an irresistible and permanent love potion.
6
{Uhura's quarters}
Kirk: My communications officer vacated these quarters for you and-
Elaan: Boo-hoo, nobody likes me!
Caption: tears!
7
{engineering}
Caption: Plot complication!
Kryton: I'm a Klingon agent! You won't escape them!
SFX: zap! suicide!
Petri: Elaan's guard has been shot! Round up the usual suspects.
8
{bridge}
Kirk: Fire!
Sulu: Klingon ship hit, captain!
Petri: I shall report it to Troyius. I haven't decided if they committed suicide or died trying to escape.
9
{transporter room}
Kirk: You're getting on that planet with the ruler of Troyius, where you belong.
Elaan: But I have chosen you!
Kirk: If you stay, we'll both wind up in a Klingon concentration camp.
Elaan: But what about us?
10
Kirk: We'll always have Uhura's quarters. But I've got a job to do. It doesn't take much to see that the problems of three people don't amount to a hill of beans in this crazy Galaxy.
Kirk: Here's looking at you, kid.
Kirk: Energise.
11
{bridge}
McCoy: How's Jim going to function with a biochemical toxin causing him to love that woman forever?
Spock: The antidote to a woman of Elas, Doctor, is a starship.
12
Kirk: How did you survive being stabbed in the heart by Elaan?
Petri: That is my least vulnerable spot.
Kirk: Petri, I think this is the beginning of a beautiful friendship.